Complex affine algebraic groups are smooth

For each $g\in G$ the left translation ${\lambda }_g:G\to G$, ${\lambda }_g(h)=gh$, is a morphism of affine algebraic sets, because it is the composite of the pairing $({\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}}_g,{\mathrm{id}}_G):G\to G\times G$ of a constant map with the identity and the multiplication morphism $G\times G\to G$, and morphisms are closed under composition; the map ${\lambda }_{g^{-1}}$ is a two-sided inverse of ${\lambda }_g$ and has the same form, so ${\lambda }_g$ is an automorphism of $G$. These automorphisms act transitively on the point set, since ${\lambda }_{y{x}^{-1}}(x)=y$ for all $x,y\in G$.

The coordinate algebra $A=\mathbb{C}[G]$ is a finitely generated reduced $\mathbb{C}$-algebra; evaluation at a point $x$ defines the maximal ideal ${\mathfrak{m}}_x\subseteq A$, the map $x\mapsto {\mathfrak{m}}_x$ is a bijection from $G$ onto the maximal ideals of $A$, and the classical local ring is the localisation ${\mathcal{O}}_{G,x}\cong A_{{\mathfrak{m}}_x}$.

Consequently $G$ is a nonempty reduced classical finite-type space, and by step 1.1 its automorphism group acts transitively on its point set; the homogeneous-regularity supplier therefore makes every point of $G$ a regular point, that is, ${\mathcal{O}}_{G,x}\cong A_{{\mathfrak{m}}_x}$ is a regular local ring for every $x\in G$.

Every irreducible component of $G$ has dimension $\dim G$, so $G$ has pure dimension $\dim G$: by $G$ being a classical variety it is Noetherian with finitely many irreducible components $G_1,\dots ,G_m$; each ${\lambda }_g$ is a homeomorphism, hence permutes the irreducible components and preserves their chain dimensions, and the translations act transitively on points, hence on components — given components $G_i,G_j$, choose $x\in G_i$ and $y\in G_j$ lying on no other component (each component has such points because it is irreducible and not contained in the finite union of the others); the automorphism ${\lambda }_{y{x}^{-1}}$ carries the component through $x$ onto the component through $y$, so ${\lambda }_{y{x}^{-1}}(G_i)=G_j$. Thus all components have one common dimension $d$, and the finite closed cover $G=G_1\cup \cdots \cup G_m$ gives $\dim G={\max }_i\dim G_i=d$; in particular every irreducible component has dimension $\dim G$.

Distinct irreducible components of $G$ are disjoint: if $x$ lay on two components, their ideals would be distinct minimal primes $P,Q\subseteq {\mathfrak{m}}_x$ by [F14]. They remain distinct after localization: for $a\in P\setminus Q$, equality of the localized primes would imply $sa\in Q$ for some $s\mathrm{\notin }{\mathfrak{m}}_x$, contradicting primality of $Q$. These localized primes remain minimal, so the local ring would have two minimal primes, whereas it is a domain by step 2.1 and [F13]. The finitely many components are therefore open and closed, and, being irreducible, are exactly the connected components. Let $C$ be the component containing $e$. For $c\in C$, translation carries the unique component through $e$ onto the unique component through $c$, so $cC=C$. Inversion and conjugation preserve $C$ because they fix $e$ and permute components. Thus $C=G^{\circ }$ is a closed normal subgroup, its cosets are the components, and $G\mathrm{/}{G}^{\circ }$ is finite.

Every local ring of $A$ is regular: for a maximal ideal $\mathfrak{m}={\mathfrak{m}}_x$ this is $A_{\mathfrak{m}}\cong {\mathcal{O}}_{G,x}$ by step 1.2 and step 2.1; for an arbitrary prime $\mathfrak{p}\subseteq A$, a proper ideal lies in a maximal ideal, say $\mathfrak{p}\subseteq \mathfrak{m}$, and $A_{\mathfrak{p}}=(A_{\mathfrak{m}}{)}_{\mathfrak{p}{A}_{\mathfrak{m}}}$ is a prime localisation of the regular local ring $A_{\mathfrak{m}}$, hence regular. Since $A$ is a finite-type algebra over the perfect field $\mathbb{C}$, the equivalence of regularity with smoothness over a perfect field makes the scheme model $\mathrm{Spec}A$ smooth over $\mathbb{C}$.

By step 2.1 every point of $G$ is a regular point of the affine algebraic set $G$ and all its local rings ${\mathcal{O}}_{G,x}$ are regular local rings; by step 3.2 the scheme model is regular and smooth over the perfect field $\mathbb{C}$, and over a perfect field classical smoothness, scheme smoothness and regularity of all local rings agree, so $G$ is smooth over $\mathbf{C}$; by step 2.2 it has pure dimension $\dim G$. This proves the lemma; the Axiom of Choice is inherited from the named suppliers.